Hamiltonian symmetries in auxiliary-field quantum Monte Carlo calculations for electronic structure

We describe how to incorporate symmetries of the Hamiltonian into auxiliary-field quantum Monte Carlo calculations (AFQMC). Focusing on the case of Abelian symmetries, we show that the computational cost of most steps of an AFQMC calculation is reduced by $N_k^{-1}$, where $N_k$ is the number of irreducible representations of the symmetry group. We apply the formalism to a molecular system as well as to several crystalline solids. In the latter case, the lattice translational group provides increasing savings as the number of k points is increased, which is important in enabling calculations that approach the thermodynamic limit. The extension to non-Abelian symmetries is briefly discussed.Comment: 13 pages, 7 figure

Efficient ab initio auxiliary-field quantum Monte Carlo calculations in Gaussian bases via low-rank tensor decomposition

We describe an algorithm to reduce the cost of auxiliary-field quantum Monte Carlo (AFQMC) calculations for the electronic structure problem. The technique uses a nested low-rank factorization of the electron repulsion integral (ERI). While the cost of conventional AFQMC calculations in Gaussian bases scales as $\mathcal{O}(N^4)$ where $N$ is the size of the basis, we show that ground-state energies can be computed through tensor decomposition with reduced memory requirements and sub-quartic scaling. The algorithm is applied to hydrogen chains and square grids, water clusters, and hexagonal BN. In all cases we observe significant memory savings and, for larger systems, reduced, sub-quartic simulation time.Comment: 14 pages, 13 figures, expanded dataset and tex

Residue functions and Extension problems

The "qualitative" extension theorem of Demailly guarantees existence of holomorphic extensions of holomorphic sections on some subvariety under certain positive-curvature assumption, but that comes without any estimate of the extensions, especially when the singular locus of the subvariety is non-empty and the holomorphic section to be extended does not vanish identically there. Residue functions are analytic functions which connect the $L^2$ norms on the subvarieties (or their singular loci) to $L^2$ norms with specific weights on the ambient space. Motivated by the conjectural "dlt extension", this note discusses the possibility of retrieving the $L^2$ estimates for the extensions in the general situation via the use of the residue functions. It is also shown in this note that the $1$-lc-measure defined via the residue function of index $1$ is indeed equal to the Ohsawa measure in the Ohsawa--Takegoshi $L^2$ extension theorem.Comment: 10 page

A new definition of analytic adjoint ideal sheaves via the residue functions of log-canonical measures

A new definition of analytic adjoint ideal sheaves for quasi-plurisubharmonic (quasi-psh) functions with only neat analytic singularities is studied and shown to admit some residue short exact sequences which are obtained by restricting sections of the newly defined adjoint ideal sheaves to some unions of $\sigma$-log-canonical ($\sigma$-lc) centres. The newly defined adjoint ideal sheaves induce naturally some residue $L^2$ norms on the unions of $\sigma$-lc centres which are invariant under log-resolutions. They can also describe unions of $\sigma$-lc centres without the need of log-resolutions even if the quasi-psh functions in question are not in a simple normal crossing configuration. This is hinting their potential use in discussing the $\sigma$-lc centres even when the quasi-psh functions in question have more general singularities. In the course of the study, a local $L^2$ extension theorem is proven, which shows that holomorphic sections on any unions of $\sigma$-lc centres can be extended holomorphically to some neighbourhood of the unions of $\sigma$-lc centres with some $L^2$ estimates. The proof does not rely on the techniques in the Ohsawa-Takegoshi-type $L^2$ extension theorems.Comment: 41 page

Linking hopping conductivity to giant dielectric permittivity in oxides

With the promise of electronics breakthrough, giant dielectric permittivity materials are under deep investigations. In most of the oxides where such behavior was observed, charged defects at interfaces are quoted for such giant behavior to occur but the underlying conduction and localization mechanisms are not well known. Comparing macroscopic dielectric relaxation to microscopic dynamics of charged defects resulting from electron paramagnetic resonance investigations we identify the actual charged defects in the case of BaTiO3 ceramics and composites. This link between the thermal activation at these two complementary scales may be extended to the numerous oxides were giant dielectric behavior was found

An application of adjoint ideal sheaves to injectivity and extension theorems

This note reviews the authors' approach to Fujino's conjecture, i.e.~the injectivity theorem for lc pairs on compact K\"ahler manifolds, via the use of adjoint ideal sheaves coupled with the associated residue computations in their previous work. Using only the techniques and results obtained from that study and under a slightly stronger positivity assumption, a ``qualitative'' extension result is obtained. Such extension result guarantees the existence of a global holomorphic extension $F$ of any holomorphic section $f$ on some $\sigma$-lc centres of the given lc pair. The extension $F$ can be shown to take values in the corresponding adjoint ideal sheaf, even though it comes without any $L^2$ estimate of $F$ in terms of $f$. Moreover, the proof invokes and implies no vanishing theorem in general.Comment: 13 page